If y=(1+(1)/(x))^x+x^(1+(1)/(x)), find (d y)/(d x) at x=1 |...

Name: Problem 74, Chapter 5: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-03-07T12:30:35-08:00
Duration: 1 min 58 s
Channel: Rukhmani Jain
Description: If y=(1+(1)/(x))^x+x^(1+(1)/(x)), find (d y)/(d x) at x=1

Key Concepts

Logarithmic Differentiation

This technique is used to differentiate functions where the variable appears in both the base and the exponent. By taking the natural logarithm of both sides of an equation, the differentiation process is simplified using the properties of logarithms, which convert exponentials into products that are easier to differentiate.

Chain Rule

The chain rule is a fundamental concept in calculus used to differentiate composite functions. In the context of functions with variable exponents or bases, the chain rule enables the differentiation by computing the derivative of an outer function multiplied by the derivative of its inner function.

Product Rule

This rule is essential when differentiating products of two or more functions. It states that the derivative of a product is the sum of the derivative of the first function times the second function plus the first function times the derivative of the second function, which is often applicable in expressions arising from logarithmic differentiation.

Differentiation of Exponential Functions

Differentiating exponential functions, especially those with variable bases and exponents, requires combining standard differentiation techniques with logarithmic methods. This concept underpins the approach for functions that cannot be differentiated by straightforward rules due to the interdependence of the base and exponent.

Evaluation of Derivatives at Specific Points

After deriving a general expression for the derivative, it is often necessary to evaluate it at a specific point. This step involves substituting the given point into the derived expression, thereby obtaining the required numerical value of the derivative at that point.

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